Let be defined by f(x)= cases g(x), & x a

MATHEMATICS · Single correct · AP · Calculus AB · Limits and Continuity · Limits and Continuity

Problem

Let

f

be defined by

f(x)=\begin{cases}g(x), & x\neq a,\\ k, & x=a,\end{cases}

where

g

is defined for all real

x

. Which of the following conditions on

k

guarantees that

f

is continuous at

x=a

? (a)

k=g(a)

(b)

k=\dfrac{\displaystyle\lim_{x\to a^-}g(x)+\displaystyle\lim_{x\to a^+}g(x)}{2}

(c)

k=\displaystyle\lim_{x\to a}g(x)

and this limit exists (d)

k=\displaystyle\lim_{x\to a^-}g(x)